1. ## Sphere Curve

Please show that the parameter curve : {$\displaystyle x=\frac{t}{1+t^2+t^4},y=\frac{t^2}{1+t^2+t^4},z=\f rac{t^3}{1+t^2+t^4}$},($\displaystyle -\infty<t<+\infty$) is a sphere curve. then write the formular of the sphere.

Thanks very much

2. Originally Posted by Xingyuan
Please show that the parameter curve : {$\displaystyle x=\frac{t}{1+t^2+t^4},y=\frac{t^2}{1+t^2+t^4},z=\f rac{t^3}{1+t^2+t^4}$},($\displaystyle -\infty<t<+\infty$) is a sphere curve. then write the formular of the sphere.

Thanks very much
let $\displaystyle u=1+t^2+t^4.$ so the curve is: $\displaystyle x=\frac{t}{u}, \ y=\frac{t^2}{u}, \ z=\frac{t^3}{u}.$ we want to find the constants $\displaystyle a,b,c,d$ such that $\displaystyle (x-a)^2+(y-b)^2+(z-c)^2=d^2,$ for all $\displaystyle t.$ i.e.

$\displaystyle (t-au)^2+(t^2-bu)^2+(t^3-cu)^2=d^2u^2.$ simplifying the LHS gives us: $\displaystyle (a^2+b^2+c^2)u^2-(2at+(2b-1)t^2+2ct^3)u=d^2u^2.$ thus: $\displaystyle a=c=0, \ \ b=d=\frac{1}{2}.$

so, just for fun, you can check my result by proving that: $\displaystyle \forall t \in \mathbb{R}: \ \ \left(\frac{t}{1+t^2+t^4} \right)^2 + \left(\frac{t^2}{1+t^2+t^4}-\frac{1}{2} \right)^2 + \left(\frac{t^3}{1+t^2+t^4} \right)^2=\frac{1}{4}. \ \ \ \Box$